The aim of this paper is to analyze a collection of data gathered from surveys held every three weeks in a Spring Course of the Economic Faculty in the University “Ismail Qemali”of Vlora, Albania. The data set for each student also contains the names of other students through which he/she have a “social relationship”. This social relationship includes frequent communications, discussions on exercise solutions, and sitting usually close to each other in the class. We have constructed four social simple graphs and have analyzed them focusing only on degrees. In addition, we fit discrete power law degree distribution on the tail and their evolution through time. In analyzing the data, we employed the R platform

Gjermeni, Orgeta

English

European Scientific Journal, ESJ

European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  129 POWER LAW DISTRIBUTION AS A COMPONENT OF THE VERTEX DEGREE DISTRIBUTION ON A SOCIAL UNIVERSITY NETWORK COURSE    Orgeta Gjermëni, PhD Candidate University “Ismail Qemali”, Vlore, Albania   Abstract   The aim of this paper is to analyze a collection of data gathered from surveys held every three weeks in a Spring Course of the Economic Faculty in the University “Ismail Qemali”of Vlora, Albania. The data set for each student also contains the names of other students through which he/she have a “social relationship”. This social relationship includes frequent communications, discussions on exercise solutions, and sitting usually close to each other in the class. We have constructed four social simple graphs and have analyzed them focusing only on degrees. In addition, we fit discrete power law degree distribution on the tail and their evolution through time. In analyzing the data, we employed the R platform.  Keywords: Social Network, Vertex Degree, Power Law  Introduction  Social network is a field which has emerged from social psychology, sociology, statistics, and graph theory since 1930’s. Jacob Moreno was the first who studied the interpersonal relationships. Mathematical formalization happened in the 1950’s while theory and methods of social networks became pervasive in the social and behavioral sciences during the 1980’s (Wasserman & Faust, 1994; Freeman, 2004). Thus, a social network is a set of people or groups of people with some pattern of contacts or interaction between them (Scott, 2000; Wasserman & Faust, 1994).  However, modern social network analysis is a computationally intensive affair nowadays. It is also related to big data. “Computational Social Science” (Lazer et al., 2009;  Kolaczyk, 2009; Kolaczyk & Csárdi, 2014)  is a new discipline, which has emerged to join the efforts of social scientists, computer scientist, mathematicians, and physicists in an interdisciplinary approach, with the purpose of better understanding the European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  130 behaviors laws of human society at both the individual and collective level. The dependence of network analysis on computations for research has helped in spawning a wide array of software packages for performing analytic tasks such as: R, Graphiz, Pajek, Cytoscape, and Gephi.  In our work, we have chosen the open statistical computing platform, R (R Core Team, 2015).    Social networks as real – world random graphs, which evolve in time, are studied mostly on static snapshots at various points in time. Subsequently, these snapshots are used to make inferences about the evolutionary processes. Hence, various studies (Newman, 2003; Guimerà et al., 2006; Kumar et al., 2006; Onnela et al., 2007) were carried out in this field.  Methods  Data was obtained from four surveys which were held every three weeks in a Spring Statistic Course for the second year students of the Economic Faculty in the University “Ismail Qemali” of  Vlora. At the beginning of each course,  students have the possibility to choose between some alternatives of the lecturer they want. A “mixing process” usually happens at the beginning of every course within the various groups of students.  The first survey was held after the course began after three weeks. During the survey, each of the students provided the names of other students which he/she has a “social relationship” with. This social relationship includes frequent communications, discussions on exercise solutions, and sitting usually close to each other in the class. Therefore, this “relationship” defines the socialization that happens within the university course. The surveys which were conducted were considered as “snapshots” at four different moments during the time of the survey. If a “social relationship” starts between two students, it is considered to be “forever” until the course ends.                   European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  131  Figure1: Visualization of evolving process in the social graph captured at four moments.   Furthermore, we will apply the mathematical definition of the network, “graph”, since analysis is focused on the mathematical context. Our social graph is conceived as a fixed set of vertices. In addition, it is seen at four moments 𝐺𝑖 = (𝑉𝑖,𝐸𝑖), where 𝑖 = 1,4���� (see Figure 1). Thus, for every 1 ≤ 𝑖 < 4, 𝐺𝑖 ⊆ 𝐺𝑖+1. In this way, we have the possibility to study the evolving process that happens between the relations held in this course. Let 𝑉𝑖 − be the set of vertices (every vertex identifies a student) which is fixed (|𝑉𝑖| =89) from the beginning of the semester, and 𝐸𝑖 − be the set of edges (every edge identifies a relationship between two students) which has the tendency to grow pass the time (|𝐸1| = 128, |𝐸2| = 289,  |𝐸3| = 414,|𝐸4| = 497). Hence, the orientation of the edges is neglected and the graphs are simplified. Degree of a vertex can be defined as the number of incident edges with it. The connectivity of a graph is related with the degree distribution (probability mass function (pmf) in the discrete distribution or probability density function (pdf) in the continuous distribution), p(x) =P(Xi = x), which measures the probability that a random vertex will have degree equal to x. The cumulative distribution function (cdf), Fi(x) =P(Xi ≤ x) measures the probability of degrees less than x. The European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  132 complementary cumulative distribution function (ccdf), Fı(x)������� = P(Xi > 𝑥), measures the probability of degrees greater than x .   During the study, we investigated the presence of a power law component on the “tail” of the degree distribution (for 𝑥 greater than a minimal value 𝑥𝑚𝑖𝑛), given that CCDF plotted on logarithmic scale on both axes has a linear decrease component on the “tail” (see Figure 3). However, we cannot be sure of the presence of a power law known as “linear component”. Therefore, what we see in Figure 3 is a necessary condition but not a sufficient one!  Vertex degree is a discrete random variable. As a result, we focus on the investigation of discrete power law. The probability mass function (pmf) of the discrete power law is expressed by the formula below: 𝑝(𝑥) = 𝐶x−α,    (1) where  ku C = 1ζ(α,xmin), is the normalization constant in way that  ∑ 𝐶 𝑝(𝑥) = 1∞x=xmin  . ζ(α, xmin) = ∑ (n + xmin)−α∞n=0 , (2)  Consequently, the above equation is the generalized function or Hurwitz zeta function (Abramowitz & Stegun, 1972). The complementary cumulative distribution function (ccdf) is given as: 𝐹(𝑥)������ = ζ(α,x)ζ(α,xmin).   (3)  Estimations on discrete parametric power law which better fits on the empirical data, were done as explained by Clauset et al. (2009). Furthermore, it was implemented on poweRlaw package (Gillespie, 2014; Gillespie, 2015). The 𝑥�𝑚𝑖𝑛 value estimated is chosen in way that the estimated power law model gets a best fit of the empirical probability distribution for  𝑥 ≥ 𝑥𝑚𝑖𝑛 (Clauset et al., 2007). Under the supposition that our data follows a power law for 𝑥 ≥ 𝑥𝑚𝑖𝑛,, the α parameter is estimated by a numeric optimization of the log – likelihood. In the discrete power law, α� is approximated withα� ≅ 1 + n �∑ 𝑙𝑛 𝑥𝑖𝑥𝑚𝑖𝑛−1/2𝑛𝑖=1 �−1. Moreover, the details of this approximation are given by Clauset et al. (2009). To estimate the distance between the two model distributions, the empirical and theoretical power law uses the Kolmogorov – Smirnov statistic (KS) (Press et al., 1992).  𝑫 = 𝑚𝑎𝑥𝑥≥𝑥𝑚𝑖𝑛�𝑆(𝑥) − 𝐹(𝑥)�������, ,  (4) where S(x) is the empirical CCDF, while 𝐹(𝑥)������ is the theoretical CCDF of the power law model which best fits the empirical data for 𝑥 ≥ 𝑥𝑚𝑖𝑛. Estimation for 𝑥�𝑚𝑖𝑛 is defined by the value which minimizes D. Uncertainty on  𝑥𝑚𝑖𝑛 estimation is investigated by the non parametric bootstrap method (Efron & Tibshirani, 1993) with 5000 iterations. To estimate  𝛼 and 𝑥𝑚𝑖𝑛 during bootstraps analyses, the maximum likelihood estimation (MLE) is used.  European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  133  For every data set, it is possible to make a power law fit, despite the fact that the real data are not following this distribution. For this reason, we need to make a test between two hypotheses:   𝐻0: data is generated according to a power law distribution for 𝑥 ≥ 𝑥𝑚𝑖𝑛.  𝐻1: data is not generated according to a power law distribution for 𝑥 ≥ 𝑥𝑚𝑖𝑛.  Hypothesis testing procedure is done as explained by Clauset et al. (2009) based on a goodness – of – fit test. This test however generates a p – value that quantifies the plausibility of the hypothesis. Such tests are based on measurements of the “distance” between the distribution of empirical data and the hypothesized model. Consequently, 𝐻0 hypothesis is ruled out for 𝑝 ≤ 0.1.   The statistical packages used for analysis include: igraph (Csárdi & Nepusz, 2006), poweRlaw (Gillespie, 2014; Gillespie, 2015), and network (Butts,  2008; Butts,  2015 ) on the R platform.    Results  To have a general overview on degree evolving process, the evolving degree diagram (see Figure 2) was constructed. Passing from G1 to G2, there is an increase by 3.629 on average to the degrees of vertices from G2 to G3 with 2.798. Furthermore, it also increases from G3 to G4 with 1.865. It is noted that over time, there is an average growth decline on degrees per vertex. After plotting all the complementary cumulative distribution functions (CCDF) on the logarithmic scale on both axes (see Figure 3), it was noted that a “linear” decline which suggests raising the hypotheses was present.   Figure 2: Evolving degree diagram for each vertex of the social graph at four moments.  In addition, a power law component is present especially on the “tail” of the degree distribution. For each of the graphs (G1, G2, G3 and G4), an European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  134 estimation of best fitted power law is done. Also, the bootstrap procedure with 5000 iterations is also applied. The results of this procedure are given in Table 1. Also, we have visualized the CCDF’s histogram of 𝑥𝑚𝑖𝑛, histogram of 𝛼, and the scatter plot of 𝑥𝑚𝑖𝑛 versus 𝛼, based on the bootstrap procedure with 5000 iterations (see Figure 4, 5, 6, and 7). Table 1: Results gained through the fitting procedure of discrete power law model to the empirical degree distribution based also on the bootstrap procedure with 5000 iterations. Distribution Graph KS 𝒙𝒎𝒊𝒏 Sd(𝒙𝒎𝒊𝒏) α Sd(𝛂 ) p -value Power law (Discrete) G1 0.0639 7 1.187 5.85 1.954 0.58 G2 0.0661 13 1.41 8.66 2.27 0.81 G3 0.0846 17 2.72 7.05 2.09 0.59 G4 0.1282 20 3.648 6.85 2.2 0.14  Figure 3: Complementary cumulative distribution functions (CCDF) given at four moments, plotted on logarithmic scale on both axes. Zero degrees are not considered.  Figure 4: a) CCDF of the data set and the fitted power law linear line of G1 with 𝑥𝑚𝑖𝑛 = 7 and parameter 𝛼 = 5.85; b) histogram of 𝑥𝑚𝑖𝑛 (sd= 1.187); c) histogram of the distribution parameter (Par 1) = 𝛼 (sd=1.954); and d) scatter plot of parameter  𝛼  versus𝑥𝑚𝑖𝑛.  European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  135  Figure 5: a) CCDF of the data set and the fitted power law linear line of G2 with 𝑥𝑚𝑖𝑛 = 13 and parameter 𝛼 = 8.66; b) histogram of 𝑥𝑚𝑖𝑛    (sd= 1.41); c) histogram of the distribution parameter (Par 1) = 𝛼 (sd=2.27); and d) scatter plot of parameter  𝛼  versus 𝑥𝑚𝑖𝑛.  Figure 6: a) CCDF of the data set and the fitted power law linear line of G3 with 𝑥𝑚𝑖𝑛 = 17 and parameter 𝛼 = 7.05; b) histogram of 𝑥𝑚𝑖𝑛 (sd= 2.72); c) histogram of the distribution parameter (Par 1) = 𝛼 (sd=2.09); and d) scatter plot of parameter  𝛼  versus 𝑥𝑚𝑖𝑛 .  Figure 7: a) CCDF of the data set and the fitted power law linear line of G4 with 𝑥𝑚𝑖𝑛 = 20 and parameter 𝛼 = 6.85; b) histogram of 𝑥𝑚𝑖𝑛   (sd= 3.648); c) histogram of the distribution parameter (Par 1) = 𝛼 (sd=2.2); and d) scatter plot of parameter  𝛼  versus 𝑥𝑚𝑖𝑛 . European Scientific Journal July 2015 edition vol.11, No.20  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431  136 Conclusion and Discussion  During this study, the student’s ability to socialize was observed. Thus, socialization among students was greater at the beginning of the course than at the end. Hypothesis raised about the presence of a power law component in degree distributions of the social graph at all the four moments, are not ruled out according to the p – values (see Table 1). Although in the case of G4, p – value (0.14) is too close to the critical value which is 0.1. For the minimal degree values 𝑥𝑚𝑖𝑛 for which it was estimated, a theoretical discrete power law distribution for the empirical data, in each of the cases have a tendency to grow passing the time. However, the estimated values of the parameter 𝛼 fluctuate with time. 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POWER LAW DISTRIBUTION AS A COMPONENT OF THE VERTEX DEGREE DISTRIBUTION ON A SOCIAL UNIVERSITY NETWORK COURSE

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POWER LAW DISTRIBUTION AS A COMPONENT OF THE VERTEX DEGREE DISTRIBUTION ON A SOCIAL UNIVERSITY NETWORK COURSE

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